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▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐✯ ❛♥❞ ❈❛r♠✐♥❡ ❱❡♥tr❡

❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦ ❯♥✐✈❡rs✐t② ♦❢ ▲✐✈❡r♣♦♦❧

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

▼♦❞❡❧

✶ ✷ ✸ ✹ ✺ ❆ ❇ ❈ ♥ = ✺, ♠ = ✸

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

▼♦❞❡❧

✶ ✷ ✸ ✹ ✺ ❆ ❇ ❈ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ♥ = ✺, ♠ = ✸

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

▼♦❞❡❧

✶ ✷ ✸ ✹ ✺ ❆ ❇ ❈

✇✶❆ = ✺ ✇✷❆ = ✸✶ ✇✷❇ = ✷✸ ✇✸❇ = ✺✺ ✇✸❈ = ✶.✺ ✇✹❈ = ✷✹ ✇

✺ ❈

= ✼

❘❡str✐❝t❡❞ ❯♥r❡❧❛t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ♥ = ✺, ♠ = ✸ ✇♠✐♥ = ✶.✺ ✇♠❛① = ✺✺ s = ✇♠❛①

✇♠✐♥ = ✺✺ ✶.✺

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

▼♦❞❡❧

✶

✇✶ = ✹

✷

✇✷ = ✻

✸

✇✸ = ✾

✹

✇✹ = ✺

✺

✇✺ = ✶

❆

✈❆ = ✷

❇

✈❇ = ✼

❈

✈❈ = ✹

❘❡str✐❝t❡❞ ❘❡❧❛t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ♥ = ✺, ♠ = ✸ ✇✐❥ =

✇✐ ✈❥

r = ♠❛①❥ ✈❥

♠✐♥❥ ✈❥ = ✼ ✷

t = ♠❛①✐ ✇✐

♠✐♥✐ ✇✐ = ✾ ✶

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

▼♦❞❡❧

✶

❥✶ = ❆

✷

❥✷ = ❇

✸

❥✸ = ❈

✹

❥✹ = ❈

✺

❥✺ = ❈

❆

♥❆ = ✶ ❝❆ = ✺

❇

♥❇ = ✶ ❝❇ = ✷✸

❈

♥❈ = ✸ ❝❈ = ✸✷.✺ ✇✶❆ = ✺ ✇✷❆ = ✸✶ ✇✷❇ = ✷✸ ✇✸❇ = ✺✺ ✇✸❈ = ✶.✺ ✇✹❈ = ✷✹ ✇

✺ ❈

= ✼

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥ ❆♥ ❖♣t✐♠❛❧ ❙♦❧✉t✐♦♥ ✭❖P❚✮ ✐s ❛♥ ❛ss✐❣♥♠❡♥t t❤❛t ♠✐♥✐♠✐③❡ t❤❡ ♦❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s❡❧❡❝t❡❞ ❜② t❤❡ ❞❡s✐❣♥❡r✳ ✭❊①✿

✐ ❝❥✐ ✮

✶

❥∗

✶ = ❆

✷

❥∗

✷ = ❇

✸

❥∗

✸ = ❈

✹

❥∗

✹ = ❈

✺

❥∗

✺ = ❈

❆

♥∗

❆ = ✶

❝∗

❆ = ✺

❇

♥∗

❇ = ✶

❝∗

❇ = ✷✸

❈

♥∗

❈ = ✸

❝∗

❈ = ✸✷.✺

✇✶❆ = ✺ ✇✷❆ = ✸✶ ✇✷❇ = ✷✸ ✇✸❇ = ✺✺ ✇✸❈ = ✶.✺ ✇✹❈ = ✷✹ ✇

✺ ❈

= ✼

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥ ❆ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ❙♦❧✉t✐♦♥ ✭◆❆❙❍✮ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ✇❤❡r❡ ♥♦ ❥♦❜ ❝❛♥ ✉♥✐❧❛t❡r❛❧❧② ✐♠♣r♦✈❡ ✐ts ❧❛t❡♥❝② ♠♦✈✐♥❣ t♦ ❛♥♦t❤❡r ♠❛❝❤✐♥❡✳ ∀ ❥♦❜ ✐, ∀ ♠❛❝❤✐♥❡ ❥, ❝❥✐ ≤ ❝❥ + ✇✐❥. ✶

❥✶ = ❆

✷

❥✷ = ❇

✸

❥✸ = ❈

✹

❥✹ = ❈

✺

❥✺ = ❈

❆

♥❆ = ✶ ❝❆ = ✺

❇

♥❇ = ✶ ❝❇ = ✷✸

❈

♥❈ = ✸ ❝❈ = ✸✷.✺ ✇✶❆ = ✺ ✇✷❆ = ✸✶ ✇✷❇ = ✷✸ ✇✸❇ = ✺✺ ✇✸❈ = ✶.✺ ✇✹❈ = ✷✹ ✇

✺ ❈

= ✼

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

Pr✐❝❡ ♦❢ ❆♥❛r❝❤②

❉❡✜♥✐t✐♦♥ ❚❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ✭P♦❆✮ ❬❑P✾✾❪ ✐s t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ❝♦st ♦❢ t❤❡ ✇♦rst ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ❙♦❧✉t✐♦♥ ❛♥❞ t❤❡ ❝♦st ♦❢ t❤❡ ❖♣t✐♠❛❧ ❙♦❧✉t✐♦♥✳ P♦❆ = ❈(◆❆❙❍) ❈(❖P❚) ■t ♠❡❛s✉r❡s ❤♦✇ ♠✉❝❤ t❤❡ ❧❛❝❦ ♦❢ ❝❡♥tr❛❧ ❝♦♦r❞✐♥❛t✐♦♥ ❝❛♥ ❛✛❡❝t t❤❡ s②st❡♠✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

Pr❡✈✐♦✉s ❘❡s✉❧ts

❊①✐st❡♥❝❡ ❆ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ❙♦❧✉t✐♦♥ ❛❧✇❛②s ❡①✐sts✳ ❬❊❞❑▼✵✸❪ ▼❛①✐♠✉♠ ▲❛t❡♥❝② (♠✐♥ ♠❛①❥ ❝❥) P♦❆ = Θ

• s +

❧♦❣ ♠ ❧♦❣(✶+ ❧♦❣ ♠

s

)

• ✐♥ t❤❡ ❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣❀ ❬❆❆❘❚✵✸❪

P♦❆ = Θ

• ❧♦❣ ♠

❧♦❣ ❧♦❣ ♠

• ✐♥ ❘❡❧❛t❡❞ ❙❡tt✐♥❣✳ ❬❆❆❘❚✵✸❪

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❖✉r ❘❡s✉❧ts

❇♦✉♥❞s ♦♥ P♦❆ ❢♦r ❙❡r✈❡r ▲❛t❡♥❝②❀ ❚♦t❛❧ ▲❛t❡♥❝②❀ ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝②

❉❡✜♥✐t✐♦♥ ❚❤✐s ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♠✐♥✐♠✐③❡s t❤❡ t♦t❛❧ ❧♦❛❞ s✉✛❡r❡❞ ❜② t❤❡ ♠❛❝❤✐♥❡s✳ ♠✐♥

❥ ❝❥ = ♠✐♥ ✐ ✇✐❥✐

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✷

❖✉r ❘❡s✉❧t ✲ ❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ P♦❆ = Θ(s) ❖✉r ❘❡s✉❧t ✲ ❘❡❧❛t❡❞ ❙❡tt✐♥❣ P♦❆ = Θ(♠✐♥(r, ♥))

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✸

❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ❯♣♣❡r ❇♦✉♥❞ ✇✐❥✐ ≤ s✇✐❥∗

✐ ⇒

❥ ❝❥ ≤ s ❥ ❝∗ ❥

❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ✶ ✷ ✸ ❆ ❇ ❈

✇✶❆ = ✶ ✇✶❇ = s ✇✷❈ = s ✇✷❇ = ✶ ✇✸❈ = ✶ ✇

✸ ❆

= s

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✸

❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ❯♣♣❡r ❇♦✉♥❞ ✇✐❥✐ ≤ s✇✐❥∗

✐ ⇒

❥ ❝❥ ≤ s ❥ ❝∗ ❥

❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ✶ ✷ ✸ ❆ ❇ ❈

✇✶❆ = ✶ ✇✶❇ = s ✇✷❈ = s ✇✷❇ = ✶ ✇✸❈ = ✶ ✇

✸ ❆

= s

❈(❖P❚) = ✸

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✸

❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ❯♣♣❡r ❇♦✉♥❞ ✇✐❥✐ ≤ s✇✐❥∗

✐ ⇒

❥ ❝❥ ≤ s ❥ ❝∗ ❥

❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ✶ ✷ ✸ ❆ ❇ ❈

✇✶❆ = ✶ ✇✶❇ = s ✇✷❈ = s ✇✷❇ = ✶ ✇✸❈ = ✶ ✇

✸ ❆

= s

❈(◆❆❙❍) = ✸s

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✹

❘❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ∀ ε > ✵ P♦❆ ≥

• r − ε

✇✐t❤ ♠✐♥(r, ♥) = r ♥ − ε ✇✐t❤ ♠✐♥(r, ♥) = ♥ ❛✶

✇❛✶ = r

❛✷

✇❛✷ = ✶

❛♥

✇❛♥ = ✶

❜✶

✈❜✶ = r

❜✷

✈❜✷ = ✶

❜♥

✈❜♥ = ✶

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✹

❘❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ∀ ε > ✵ P♦❆ ≥

• r − ε

✇✐t❤ ♠✐♥(r, ♥) = r ♥ − ε ✇✐t❤ ♠✐♥(r, ♥) = ♥ ❛✶

✇❛✶ = r

❛✷

✇❛✷ = ✶

❛♥

✇❛♥ = ✶

❜✶

✈❜✶ = r

❜✷

✈❜✷ = ✶

❜♥

✈❜♥ = ✶

❈(❖P❚) = r+♥−✶

r

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✹

❘❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ∀ ε > ✵ P♦❆ ≥

• r − ε

✇✐t❤ ♠✐♥(r, ♥) = r ♥ − ε ✇✐t❤ ♠✐♥(r, ♥) = ♥ ❛✶

✇❛✶ = r

❛✷

✇❛✷ = ✶

❛♥

✇❛♥ = ✶

❜✶

✈❜✶ = r

❜✷

✈❜✷ = ✶

❜♥

✈❜♥ = ✶

❈(◆❆❙❍) = ♥

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❙❡r✈❡r ▲❛t❡♥❝② ✲ ✺

❘❡❧❛t❡❞ ❙❡tt✐♥❣ ✲ ▲♦✇❡r ❇♦✉♥❞ ∀ ε > ✵ P♦❆ ≥

• r − ε

✇✐t❤ ♠✐♥(r, ♥) = r ♥ − ε ✇✐t❤ ♠✐♥(r, ♥) = ♥ ▲❡t ♥ ≥ r ✷

ε ✱ ❛♥❞ s♦ ♠✐♥(♥, r) = r✱ t❤❡♥ P♦❆ ≥ r − ε❀

▲❡t r ≥ ♥✷

ε ✱ ❛♥❞ s♦ ♠✐♥(♥, r) = ♥✱ t❤❡♥ P♦❆ ≥ ♥ − ε✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝②

❉❡✜♥✐t✐♦♥ ❚❤✐s ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♠✐♥✐♠✐③❡s t❤❡ t♦t❛❧ ❧❛t❡♥❝② s✉✛❡r❡❞ ❜② ❥♦❜s✳ ♠✐♥

✐ ❝❥✐ = ♠✐♥ ❥ ♥❥❝❥

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝② ✲ ✷

Pr❡✈✐♦✉s ❘❡s✉❧ts ✲ ❯♥r❡str✐❝t❡❞ ❘❡❧❛t❡❞ ❙❡tt✐♥❣ P♦❆ ≤ ✹s❀ ❬❇●●▼✵✻❪

♥ ✷✇ ≤ P♦❆ ≤ ♥ ✇ + ♠✷+♠ ✇ ✷ ✳ ❬❍❙✵✼❪

Pr❡✈✐♦✉s ❘❡s✉❧ts ✲ ❘❡str✐❝t❡❞ ❯♥✇❡✐❣❤t❡❞ ❘❡❧❛t❡❞ ❙❡tt✐♥❣ ✷.✺ − ε ≤ P♦❆ ≤ ✷.✺✳ ❬❙❚❩✵✹❪❬❈❋❑+✵✻❪ ❖✉r ❘❡s✉❧ts ✲ ❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ P♦❆ = ❖(♠s)❀ P♦❆ = Ω(s)✳ ❖✉r ❘❡s✉❧ts ✲ ❘❡❧❛t❡❞ ❙❡tt✐♥❣ P♦❆ = Ω

• ♠✐♥
• ♠+√t

♠+✸ , ♥ ♠+✸

• ✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝② ✲ ✸

❯♣♣❡r ❇♦✉♥❞ ∀ ❥♦❜ ✐, ❝❥✐ ≤ ❝❥∗

✐ + ✇✐❥∗ ✐

• ✐

❝❥✐ ≤

• ✐

(❝❥∗

✐ + ✇✐❥∗ ✐ ) ≤

• ✐

(❝❥∗

✐ + ❝∗

❥∗

✐ )

❈(◆❆❙❍) =

• ❥

♥❥❝❥ ≤

• ❥

♥∗

❥ ❝❥ +

• ❥

♥∗

❥ ❝∗ ❥ =

• ❥:❝∗

❥ =✵

♥∗

❥ ❝❥ +❈(❖P❚) ✭✶✮

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝② ✲ ✹

❯♣♣❡r ❇♦✉♥❞ ◆♦✇ ✇❡ ❝❛♥ ✉♣♣❡r✲❜♦✉♥❞

❥ ♥∗ ❥ ❝❥ ❛s ❢♦❧❧♦✇s✿

• ❥

♥∗

❥ ❝❥ ≤

• ❥

♥∗

❥ ♥❥✇♠❛① ≤ ✇♠❛①(

• ❥

♥∗

❥ )(

• ❥

♥❥) = ♥✷✇♠❛① ✭✷✮ ❲❡ ❝❛♥ ❡❛s✐❧② ❧♦✇❡r✲❜♦✉♥❞ ❈(❖P❚) ❛s ❢♦❧❧♦✇s✿

• ❥

♥∗

❥ ❝∗ ❥ ≥

• ❥
• ♥∗

❥

✷ ✇♠✐♥ ≥ ✇♠✐♥

• ❥ ♥∗

❥

✷ ♠ = ♥✷ ♠ ✇♠✐♥ ✭✸✮ ❋r♦♠ ✭✶✮✱ ✭✷✮ ❛♥❞ ✭✸✮ ✇❡ ♦❜t❛✐♥ P♦❆ = ❈(◆❆❙❍) ❈(❖P❚) ≤ ✶ +

• ❥ ♥∗

❥ ❝❥

❈(❖P❚) ≤ ✶ + ♠♥✷✇♠❛① ♥✷✇♠✐♥ = ❖(♠s).

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝② ✲ ✺

▲♦✇❡r ❇♦✉♥❞ ✲ ❘❡❧❛t❡❞ ❙❡tt✐♥❣

❆✶ |❆✶| = ⌈r⌉✇✱ ✇❆✶ = ✶ ❆✷ |❆✷| = ⌈r⌉✇✱ ✇❆✷ = ✶ ❝✶ ❝✷ ✇❝✐ = t ≥ (✷⌈r⌉✇ + ❧✇)✷ ❇✶ ❇✷ ❇♠−✷ |❇✐ | = ✇✱ ✇❇✐ = ✶ ❞✶ ❞✷ ❞❧ ❧ = ♠ − ✷✱ ✇❞✐ = t r ❢✶ ❢✷ ✈❢✶ = ✈❢✷ = r ❢✸ ❢♠ ✈❢✐ = ✶ ✇ = ‰ (❦−✶)(♠+✸)+✸ ✷r+♠−✷ ı ❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝② ✲ ✺

▲♦✇❡r ❇♦✉♥❞ ✲ ❘❡❧❛t❡❞ ❙❡tt✐♥❣

❆✶ |❆✶| = ⌈r⌉✇✱ ✇❆✶ = ✶ ❆✷ |❆✷| = ⌈r⌉✇✱ ✇❆✷ = ✶ ❝✶ ❝✷ ✇❝✐ = t ≥ (✷⌈r⌉✇ + ❧✇)✷ ❇✶ ❇✷ ❇♠−✷ |❇✐ | = ✇✱ ✇❇✐ = ✶ ❞✶ ❞✷ ❞❧ ❧ = ♠ − ✷✱ ✇❞✐ = t r ❢✶ ❢✷ ✈❢✶ = ✈❢✷ = r ❢✸ ❢♠ ✈❢✐ = ✶ ✇ = ‰ (❦−✶)(♠+✸)+✸ ✷r+♠−✷ ı ❈(❖P❚) ≤ (♠ + ✸) t r ❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❚♦t❛❧ ▲❛t❡♥❝② ✲ ✺

▲♦✇❡r ❇♦✉♥❞ ✲ ❘❡❧❛t❡❞ ❙❡tt✐♥❣

❆✶ |❆✶| = ⌈r⌉✇✱ ✇❆✶ = ✶ ❆✷ |❆✷| = ⌈r⌉✇✱ ✇❆✷ = ✶ ❝✶ ❝✷ ✇❝✐ = t ≥ (✷⌈r⌉✇ + ❧✇)✷ ❇✶ ❇✷ ❇♠−✷ |❇✐ | = ✇✱ ✇❇✐ = ✶ ❞✶ ❞✷ ❞❧ ❧ = ♠ − ✷✱ ✇❞✐ = t r ❢✶ ❢✷ ✈❢✶ = ✈❢✷ = r ❢✸ ❢♠ ✈❢✐ = ✶ ✇ = ‰ (❦−✶)(♠+✸)+✸ ✷r+♠−✷ ı ❦ = ♠✐♥ „ ♠+√t ♠+✸ , ♥ ♠+✸ « ❈(◆❆❙❍) ≥ ❦(♠ + ✸) t r ❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❉❡✜♥✐t✐♦♥ ❚❤✐s ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ♠✐♥✐♠✐③❡s t❤❡ ❧❛t❡♥❝② s✉✛❡r❡❞ ❜② ❛♥② ✉♥✐t ♦❢ ❥♦❜s✬ ✇❡✐❣❤t✳ ♠✐♥

✐ ✇✐❥✐ ❝❥✐ = ♠✐♥ ❥ ❝✷ ❥

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝② ✲ ✷

Pr❡✈✐♦✉s ❘❡s✉❧t ✲ ❘❡❧❛t❡❞ ❙❡tt✐♥❣ ✷.✺ ≤ P♦❆ ≤ ✸+

√ ✺ ✷

≈ ✷.✻✶✽ ❬❆❆❊✵✺❪ ❬❈❋❑+✵✻❪ ❖✉r ❘❡s✉❧t ✲ ❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣ P♦❆ = Θ(s✷)

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝② ✲ ✸

❯♣♣❡r ❇♦✉♥❞ ∀ ❥♦❜ ✐, ❝❥✐ ≤ ❝❥∗

✐ + ✇✐❥∗ ✐ ⇒ ✇✐❥✐ ❝❥✐ ≤ ✇✐❥✐ (❝❥∗ ✐ + ✇✐❥∗ ✐ )

• ✐

✇✐❥✐ ❝❥✐ ≤

• ✐

✇✐❥✐ (❝❥∗

✐ + ✇✐❥∗ ✐ ) ≤

≤

• ✐

✇✐❥✐ (❝❥∗

✐ + ❝∗

❥∗

✐ ) ≤ s

• ✐

✇✐❥∗

✐ (❝❥∗ ✐ + ❝∗

❥∗

✐ )

❈(◆❆❙❍) =

• ❥

❝✷

❥ ≤ s(

• ❥

❝∗

❥ ❝❥ +

• ❥

(❝∗

❥ )✷) = s(

• ❥

❝∗

❥ ❝❥ + ❈(❖P❚))

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts ❙❡r✈❡r ▲❛t❡♥❝② ❚♦t❛❧ ▲❛t❡♥❝② ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝②

❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝② ✲ ✹

❯s✐♥❣ t❤❡ ✏❈❛✉❝❤②✲❙❝❤✇❛rt③ ✐♥❡q✉❛❧✐t②✑✿

• ❥

❝❥❝∗

❥ ≤

• ❈(◆❆❙❍) ·
• ❈(❖P❚)

▼❡r❣✐♥❣ t❤✐s r❡s✉❧t ✐♥ ♣r❡✈✐♦✉s st❛t❡♠❡♥t ❈(◆❆❙❍) ≤ s

• ❈(◆❆❙❍)
• ❈(❖P❚)
• + ❈(❖P❚)
• ⇒

⇒ P♦❆ − s √ P♦❆ − s ≤ ✵ ❙♦❧✈✐♥❣ t❤✐s ❞✐s❡q✉❛t✐♦♥ P♦❆ = ❖(s✷).

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

❈♦♥❝❧✉s✐♦♥s

❙❡r✈❡r ▲❛t❡♥❝② P♦❆ = Θ(s) ✐♥ ❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣❀ P♦❆ = Θ(♠✐♥(r, ♥)) ✐♥ ❘❡❧❛t❡❞ ❙❡tt✐♥❣✳ ❚♦t❛❧ ▲❛t❡♥❝② P♦❆ = ❖ (♠s)❀ P♦❆ = Ω(s) ✐♥ ❯♥r❡❧❛t❡❞ ❙❡tt✐♥❣❀ P♦❆ = Ω

• ♠✐♥
• ♠+√t

♠+✸ , ♥ ♠+✸

• ✐♥ ❘❡❧❛t❡❞ ❙❡tt✐♥❣✳

❖♣❡♥ ♣r♦❜❧❡♠✿ ❈❧♦s❡ t❤❡ ❣❛♣✳ ❲❡✐❣❤t❡❞ ❚♦t❛❧ ▲❛t❡♥❝② P♦❆ = Θ(s✷)✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

❇✐❜❧✐♦❣r❛♣❤② ■

❇❛r✉❝❤ ❆✇❡r❜✉❝❤✱ ❨♦ss✐ ❆③❛r✱ ❛♥❞ ❆♠✐r ❊♣st❡✐♥✳ ❚❤❡ ♣r✐❝❡ ♦❢ r♦✉t✐♥❣ ✉♥s♣❧✐tt❛❜❧❡ ✢♦✇✳ ■♥ ❙❚❖❈ ✬✵✺✿ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ t❤✐rt②✲s❡✈❡♥t❤ ❛♥♥✉❛❧ ❆❈▼ s②♠♣♦s✐✉♠ ♦♥ ❚❤❡♦r② ♦❢ ❝♦♠♣✉t✐♥❣✱ ♣❛❣❡s ✺✼✕✻✻✱ ✷✵✵✺✳ ❇❛r✉❝❤ ❆✇❡r❜✉❝❤✱ ❨♦ss✐ ❆③❛r✱ ❨♦ss✐ ❘✐❝❤t❡r✱ ❛♥❞ ❉❡❦❡❧ ❚s✉r✳ ❚r❛❞❡♦✛s ✐♥ ✇♦rst✲❝❛s❡ ❡q✉✐❧✐❜r✐❛✳ ■♥ ❲❆❖❆✱ ♣❛❣❡s ✹✶✕✺✷✱ ✷✵✵✸✳ P❡tr❛ ❇❡r❡♥❜r✐♥❦✱ ▲❡s❧✐❡ ❆♥♥ ●♦❧❞❜❡r❣✱ P❛✉❧ ❲✳ ●♦❧❞❜❡r❣✱ ❛♥❞ ❘✉ss❡❧❧ ▼❛rt✐♥✳ ❯t✐❧✐t❛r✐❛♥ r❡s♦✉r❝❡ ❛ss✐❣♥♠❡♥t✳ ❏♦✉r♥❛❧ ♦❢ ❉✐s❝r❡t❡ ❆❧❣♦r✐t❤♠s✱ ✹✱ ♣❛❣❡s ✺✻✼✕✺✽✼✱ ✷✵✵✻✳ ■♦❛♥♥✐s ❈❛r❛❣✐❛♥♥✐s✱ ▼✐❝❤❡❧❡ ❋❧❛♠♠✐♥✐✱ ❈❤r✐st♦s ❑❛❦❧❛♠❛♥✐s✱ P❛♥❛❣✐♦t✐s ❑❛♥❡❧❧♦♣♦✉❧♦s✱ ❛♥❞ ▲✉❝❛ ▼♦s❝❛r❞❡❧❧✐✳ ❚✐❣❤t ❜♦✉♥❞s ❢♦r s❡❧✜s❤ ❛♥❞ ❣r❡❡❞② ❧♦❛❞ ❜❛❧❛♥❝✐♥❣✳ ■♥ ■❈❆▲P ✭✶✮✱ ♣❛❣❡s ✸✶✶✕✸✷✷✱ ✷✵✵✻✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s

▼♦❞❡❧ Pr❡✈✐♦✉s ❘❡s✉❧ts ❖✉r ❘❡s✉❧ts

❇✐❜❧✐♦❣r❛♣❤② ■■

❊②❛❧ ❊✈❡♥✲❞❛r✱ ❆❧❡① ❑❡ss❡❧♠❛♥✱ ❛♥❞ ❨✐s❤❛② ▼❛♥s♦✉r✳ ❈♦♥✈❡r❣❡♥❝❡ t✐♠❡ t♦ ♥❛s❤ ❡q✉✐❧✐❜r✐❛✳ ■♥ ■❈❆▲P✱ ♣❛❣❡s ✺✵✷✕✺✶✸✱ ✷✵✵✸✳ ▼❛rt✐♥ ❍♦❡❢❡r ❛♥❞ ❆❧❡①❛♥❞❡r ❙♦✉③❛✳ ❚r❛❞❡♦✛s ❛♥❞ ❛✈❡r❛❣❡✲❝❛s❡ ❡q✉✐❧✐❜r✐❛ ✐♥ s❡❧✜s❤ r♦✉t✐♥❣✳ ■♥ ❊❙❆✱ ♣❛❣❡s ✻✸✕✼✹✱ ✷✵✵✼✳ ❊❧✐❛s ❑♦✉ts♦✉♣✐❛s ❛♥❞ ❈❤r✐st♦s P❛♣❛❞✐♠✐tr✐♦✉✳ ❲♦rst✲❝❛s❡ ❡q✉✐❧✐❜r✐❛✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✶✻t❤ ❆♥♥✉❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ❚❤❡♦r❡t✐❝❛❧ ❆s♣❡❝ts ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✹✵✹✕✹✶✸✱ ✶✾✾✾✳ ❙✉❜❤❛s❤ ❙✉r✐✱ ❈s❛❜❛ ❉✳ ❚ót❤✱ ❛♥❞ ❨✉♥❤♦♥❣ ❩❤♦✉✳ ❙❡❧✜s❤ ❧♦❛❞ ❜❛❧❛♥❝✐♥❣ ❛♥❞ ❛t♦♠✐❝ ❝♦♥❣❡st✐♦♥ ❣❛♠❡s✳ ■♥ ❙P❆❆✱ ♣❛❣❡s ✶✽✽✕✶✾✺✱ ✷✵✵✹✳

❉✐♦❞❛t♦ ❋❡rr❛✐♦❧✐ ❖♥ t❤❡ Pr✐❝❡ ♦❢ ❆♥❛r❝❤② ♦❢ ❘❡str✐❝t❡❞ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣ ●❛♠❡s